arXiv:1012.3566v2 [hep-ph] 2 Feb 2011
Signals of additional Z
′boson in e
+e
−→ W
+W
−at the ILC with polarized beams
B. Ananthanarayan Monalisa Patra Centre for High Energy Physics
Indian Institute of Science Bangalore 560 012, India
P. Poulose Department of Physics
Indian Institute of Technology Guwahati Assam 781 039, India
Abstract
We consider the possibility of fingerprinting the presence of heavy additional Z′ bosons that arise naturally in extensions of the standard model such asE6 models and left-right symmetric models, through their mixing with the standard modelZ boson. By considering a class of ob- servables including total cross sections, energy distributions and angular distributions of decay leptons we find significant deviation from the stan- dard model predictions for these quantities with right-handed electrons and left-handed positrons at√s=800 GeV. The deviations being less pro- nounced at smaller centre of mass energies as the models are already tightly constrained. Our work suggests that the ILC should have a strong beam polarization physics program particularly with these configurations.
On the other hand, a forward backward asymmetry and lepton fraction in the backward direction are more sensitive to new physics with realis- tic polarization due to interesting interplay with the neutrino t- channel diagram. This process complements the study of fermion pair produc- tion processes that have been considered for discrimination between these models.
1
1 Introduction
The International Linear Collider (ILC) is a proposed high energy, high luminos- ity electron-positron collider with the mission of studying the standard model (SM) at high precision and to look for signals beyond the standard model [1].
It has been proposed that an initial beam polarization program can also signif- icantly enhance its capabilities in meeting these objectives, see ref. [2].
One of the important processes that will be studied at high precision at ILC with and without beam polarization [2] isW-pair production. Phenomenological studies of this process within the SM and some extensions have been carried out in great detail [3, 4] starting many years ago. Since properties of the weak gauge bosons are closely linked to electroweak symmetry breaking (EWSB) and the structure of the gauge sector in general, detailed study ofW physics will throw light on what lies beyond the SM.
On the other hand, it is entirely likely that there are additionalZ bosons, denoted byZ′ in the TeV range, see for instance the review section in ref. [5].
These are present in several economical extensions of the standard model. With this strong motivation, signatures of such a gauge boson is searched for in the past and existing colliders. Direct and indirect searches at LEP as well as at TeVatron and other existing facilities provide bounds on the masses of this particle and on other model parameters. Direct searches at TeVatron put lower limits of 630−1030 GeV [6, 7] and LEP 2 put limits of 673−1787 GeV [8], while electroweak precision analysis of LEP provide lower limits of 475−1500 GeV [9]
on the mass, depending on the model considered. Even if not directly produced, they can be finger printed easily as they would mix with the traditionalZ0 of the electroweak model. Thus W-pair production process has a winning edge compared to fermion pair production when it comes to the effect of this mixing.
This is because, as these Z′ do not interact with the standardW bosons, the W-pair production process is insensitive to the presence ofZ′ in the absence of mixing. In contrast, fermion pair production process is sensitive to the presence ofZ′ even in the absence of mixing.
We note that although such mixing is highly constrained by precision mea- surements at LEP and by other existing experimental data [5], with the high statistics expected at the ILC for W-pair production, even such small mixing can be probed effectively. The new effects could be manifested in departures from the standard model cross section for W-pair production, and in various differential distributions and asymmetries.
Recently in the context of the littlest Higgs model(LHM), which also con- tain Z′ bosons, we demonstrated the utility of several simple distributions in fingerprinting the model [10]. As a result, it may be worthwhile considering how other popularZ′ models arising inE6 unification and so called left-right sym- metric models (LRSM) and alternative left-right symmetric models (ALRSM), which have also been considered by the CLIC Physics Working Group [11], can subject themselves to a diagnosis. This is the main aim of the present work.
We have studied the process at reference energies of 500, 800 and 1000 GeV, and find that effects are pronounced only at the higher energy. In addition, we
conclude that initial beam polarization can significantly enhance the diagnostic ability of the ILC. In particular, for the class of models we are considering these effects are of importance for right-handed electrons and left-handed positrons in the process we are considering. This may be understood as arising from the dominantt-channel SM contribution which gets erased away with this choice of beam polarization revealing the new physics contributing through thes-channel.
The above argument holds in the case for realistic degrees of polarization of the beams; it is expected that about 90% electron polarization would be achievable along with a positron polarization of 60% [2]. We present our results for both these cases and find that there are interesting effects even for the latter case as the t- channel contribution which now survives, can play an interesting and effective role.
We will use the observables considered by us in our earlier work, ref. [10], which are total cross sections, single energy distribution of the secondary lep- ton, lepton angular distribution and forward-backward (FB) and left-right (LR) asymmetries. In addition to the above we consider an important and useful energy-energy correlation of the type first considered in the context of some anomalous gauge couplings by Dicus and Kallianpur [12]. Despite its obvious utility it has not received much attention, and we will demonstrate how this correlation in combination with beam polarizations can extract important in- formation onZ′ models.
The scheme of this paper is the following: In Sec. 2 we discuss the details of the models we are considering. In Sec. 3 we consider the kinematics of the process and the subsequent decays in great detail. The section is organized in several subsections for convenience. In Sec. 4 we present a discussion including a comparison ofW-pair production with fermion pair production considered in the literature, and our conclusions.
2 Z
′Models
The presence of an additional neutral gauge boson (Z′) is anticipated in many extensions of the SM. Some grand unified theories (GUT) like those based on E6, which contain the gauge group ofSU(3)C×SU(2)L×U(1)Y×U(1)′clearly have additionalU(1)′ symmetries (for a recent review, see, e.g. ref. [13]). These could also arise in superstring theories. There are many other extensions of the SM with dynamical symmetry breaking, Little Higgs models (LHM), LRSM and ALRSM also with extended gauge sectors. In models with large extra dimensions, Kaluza-Klein excitations of the SM gauge bosons propagating in the bulk manifest as extra gauge bosons in four dimensions. For our purposes we confine our attention to certain GUT models based on E6 and LRSM and ALRSM. We will study the presence of an extra U(1)′ symmetry present in addition to the SM gauge symmetries, arising in the candidate models mentioned above. In some of these models there could be more than one neutral gauge bosons along with possible presence of heavy charged gauge bosons. We assume that any such additional gauge bosons decouple from the particle spectrum
3
under study, and therefore will be ignored in this study.
The new gauge boson Z′ could mix with the SM gauge boson to give the physical eigenstates. As explained later in this section, theW-pair production in e+e− collisions has the advantage of directly probing the mixing unlike, for example, the fermion pair production process. This is because, theW does not interact directly with theZ′. While such mixing is highly constrained by the current experimental constraints, it may still be possible to probe its effect in a high energy, high luminosity machine like the ILC. With very high statistics expected forW-pair production at ILC, this process has the potential to probe even very small mixing permitted.
Let us now turn to some general features of the scenario. Here we closely follow the discussion in ref. [13, 14]. With one additionalU(1)′ symmetry, the mass term of the neutral gauge bosons may be written as,
LmassZ = 1
2 Z0µ Z′µ MZ20 ∆2
∆2 MZ2′
! Zµ0 Zµ′
(1) Diagonalizing the above mass matrix the mass term is presented in terms of the physical boson fields as
LmassZ =1
2M12 Z1µZ1µ+1
2M22 Z2µZ2µ, (2) where we identify the lighterZ1as the observedZ-boson withM1= 91.19 GeV, and theZ2as its heavier counterpart. In terms of the SM gauge boson, Z0and the new gauge bosons,Z′, we may write the physical states as
Z1
Z2
=
cosθ sinθ
−sinθ cosθ
Z0 Z′
. (3)
The mixing angle, θ is related to the diagonalization of the mass matrix, and can be expressed in terms of the physical masses and the SM mass parameter as
tan2θ=MZ20−M12
M22−MZ20
. (4)
For our phenomenological analysis it is more convenient to re-parametrize the above by defining a mass difference ∆M =M1−MZ0. Rearranging Eq. 4 the mass of the heavier gauge boson takes the form
M22=(1 + tan2θ)(M1−∆M)2−M12
tan2θ . (5)
Thus, in our study we considerθand ∆M as the two independent parameters of the mixing. The importance of mixing inW-pair production at ILC is clearly visible with the fact that the new gauge sector does not interact with the SM gauge sector directly, and therefore theWboson does not couple directly toZ′at tree level. Its couplings to the mass eigenstatesZ2arises through mixing, along
with a corresponding weakening in its coupling to the lighter mass eigenstate, Z1. These couplings are given by
gW W Z1 =gW W Z0 cosθ, gW W Z2 =gW W Z0 sinθ, (6) where gW W Z0 is the SM coupling. Experimental constraints on the mixing limits the value ofθ to be not larger than a few times 10−3. At the same time, when the Higgs structure of the model is known, one may compute the mixing angle. In such a case, mixing angle can be expressed as
θ=C r5
3λsinθW
MZ21
MZ22, (7)
whereλis a parameter of order unity,C is function of the VEV’s of the Higgs fields and their U(1)′ charges andθW is the usual Weinberg mixing angle. In Table 1 we present an illustrative case ofE6models as discussed in Ref.[9]. The table also gives the mixing angle, θ and ∆M in each case corresponding to a representative value ofMZ2 = 1 TeV.
Model C (range) θ ∆M (MeV)
E6(χ) h
−√310,√210i
[−.0037, .0025] [75,33]
E6(ψ) h
−q
2 3,+q
2 3
i [−.0032, .0032] 56 E6(η) h
−√115,+√4 15
i [−.0010, .0040] [6,89]
LRSM h
−αLR1
q3 5,+αLR
q3 5
i [−.0019, .0048] [20,124]
Table 1: Mixing angle (θ) and ∆M corresponding to a Z2 of mass 1 TeV in different models considered. The parameter in the left-right symmetric model (LRSM) takes a value,αLR=p
1−2 sin2θW/sinθW.
We now turn our attention to the Zee coupling. The fermion couplings are highly model dependent. In the following we will very briefly describe this coupling in the models considered. A detailed analysis can be found in the literature including [13]. The neutral current interactions of electrons with the gauge bosons are given by the Lagrangian term,
LN C =Jemµ Aµ+JSMµ Zµ0+J′µZµ′, (8) whereJemµ is the electromagnetic current with which the photon interact,JSMµ is the current with which the SM neutral gauge boson,Z0interact, and J′µ is the current with which the new neutral gauge boson, Z′ interact. In terms of the projection operatorsPL,R= (1∓γ5)/2 the currents take the form
Jiµ =−ψ¯eγµ(geiLPL+geiRPR)ψe. (9) 5
The SM couplings involved in JSMµ are expressed in terms of the Weinberg mixing angle,θW as
gfL= e sinθWcosθW
(−1
2+ sin2θW), gRf = esinθW
cosθW
. (10) Fermion couplings corresponding to the current, J′µ in the cases of models considered in our study are tabulated in Table 2.
Model gL′e gR′e
E6(χ) q
3 8
e cosθW
q1 24
e cosθW
E6(ψ) 12 cos√10θeW −12 cos√10θeW
E6(η) 6 coseθW 3 coseθW
LRSM coseθW2α1LR coseθW(2α1LR −αLR2 ) ALRSM cosθWesin2θW
1
αLR(−12+ sin2θW) cosθWesin2θW
1
αLR(−21+32sin2θW) Table 2: Z′eecouplings (Eq. 9) in different models considered. The parameter αLR=p
1−2 sin2θW/sinθW.
In the next section, we will compare the effects found in the above models, with those arising in the Littlest Higgs scenario whose collider signature at the ILC with polarized beams was recently considered in ref. [10]. Briefly stated, in this scenario, the Higgs fields are considered to be the Nambu-Goldstone Bosons (NGB) of the non-linear realization of some global symmetry breaking.
In our numerical analysis for LHM, there are two free parametersf andθH. As argued by [15], precision electroweak measurements restrict the parameters to bef ∼1 TeV and 0.1<cosθH <0.9. In our study we consider a value of f = 1 TeV and cosθH= 0.3 satisfying these restrictions.
3 Analyses of e
+e
−→ W W
In this section we present the results of our numerical analysis to probe the Z′ models through the processe+e− →W W at the ILC. This process gets an additional s-channel contribution through the exchange of the heavier gauge bosons,Z2. Only the SM component ofZ2interacts withW, and therefore this additionals-channel contribution is not present in the absence of mixing. At the same time, Z1 becomes the SM gauge boson,Z0 with the standard couplings, when there is no mixing. Thet-channel withν-exchange is not affected by new physics here as W interactions with SM fermions proceed through standard couplings. Thus thee+e−→W W process directly probes theZ0−Z′ mixing,
unlike processes likee+e− →ff, which gets contribution from the presence of¯ Z′ even when there is no mixing.
Thus for the process e+e− → W+W−, we have an s-channel process with the exchange of the heavy neutral gauge boson,Z2, in addition to the standard channels as shown in Fig. 1.
e− e+
ν W−
W+
e−
e+
γ, Z1, Z2 W− W+
Figure 1: Feynman diagrams contributing to the process e+e− →W+W− in theZ′ models.
The three-point gauge couplings involvingW W are given by:
Vµ(k1)Wν(k2)Wρ(k3) =igV W W [gµν(k1−k2)ρ+gνρ(k2−k3)µ+gρµ(k3−k1)ν], where all the momenta are considered outflowing, andV ≡γ, Z1, Z2.
Direct search results from p¯p collisions at TeVatron as well as precision electroweak analysis constrain the parameters discussed in the previous section [5]. In most cases Z2 mass slightly smaller than 1 TeV is permitted. This is translated into allowed range of couplings in a specific version of E6 model as presented in Table 1. In our numerical studies we consider a conservative value of 0.003 and 100 MeV for the mixing angle and the parameter ∆M, respectively.
The above parameter choices correspond to aZ2of mass∼1400 GeV . For a non-exhaustive list of phenomenological studies ofZ′ in the context of LHC as well as ILC, see reference in [16, 17, 18, 19]. Previous phenomenological studies ofe+e−→W W process in the context of Z′ have considered different observables at the production level [14, 19], and have obtained the reach of ILC in probing the model. In this work our main focus is on various observables con- structed with the decay products of theW’s produced. One obvious advantage is that these observables do not require full reconstruction of theW W events, unlike in the former case. In this section, we will first report our analysis of the process at the production level with a study of the total cross section, and subsequently analyze different decay distributions and other observables.
3.1 The total cross section
We compute the total cross section incorporating beam polarization using the helicity amplitudes given in ref. [3] with the new couplings and with the added contribution due to the exchange of Z2. With beam polarization, in general, the polarized cross section may be expressed as:
7
σ(e+e− →W+W−) = 1 4
(1 +Pe−).(1−Pe+)σRL + (1−Pe−).(1 +Pe+)σLR
, (11) where σRL = σ(e+Le−R → W+W−) and σLR = σ(e+Re−L → W+W−), with eL,Rrepresenting the left- and right-polarized electrons (and positrons), respec- tively. The degree of polarization is defined as: Pe= (NR−NL)/(NR+NL), where NL,R denote the number of left-polarized and right-polarized electrons (and positrons), respectively. More than 80% of electron beam polarization and large positron beam polarization are expected to be achieved at ILC. In our analysis we consider both the ideal possibility of 100% polarization along with the realistic polarization of the beams that will be achieved at ILC.
The models considered here are insensitive to the total cross section even at higher centre of mass energies. The inclusion of beam polarizations has no significant effect. In Table 3, we present the total production cross section for SM andZ′ models considering both unpolarized and polarized beams. It can be seen that at √
s=500 GeV, the deviation for different Z′ models from the SM using unpolarized beams is about 0.05 - 0.3%. The deviation is slightly increased to about 0.7% at√
s=800 GeV. Switching on the polarization with left-handed electrons and right-handed positrons it is seen that with this choice of polarization the cross section is about four times more than the unpolarized case. Although the deviation in this case follows the same pattern as the unpo- larized one, but due to the larger size of the cross sectionZ−Z′ mixing can be studied more effectively. However with right-handed electrons and left-handed positrons, due to the absence of the dominant t channel the mixing effect is more pronounced, even though the cross section is very small compared to the other two cases. This particular combination thus leads to an increased signal by background ratio. It can be seen that the deviation is about 24% for different E6 models and 50% for the LRSM and ALRSM model at √s=800 GeV, with 100% beam polarization. We have only presented the figure for this particu- lar case. In Fig. 2 we present the total production cross-section in the case of SM and E6(χ) as this has a maximum deviation compared to other E6 mod- els. Moreover, it can be seen that the percent deviation of LRSM and ALRSM are equal with this specific choice of beam polarization. LHM is also plotted for comparison. Since the gauge structure of LHM is different from the other Z′ models considered here, it behaves differently from them. It is not much constrained compared to the other models as can be seen from Fig. 2. Taking into account the polarization which will be achieved at ILC, differentE6models show about 4% deviation from the SM. The percent deviation for LRSM and ALRSM is about 9% compared to 50% obtained in the ideal case.
3.2 Double Energy Distribution
In order to exploit further the process at hand, it is profitable to consider the decays of one or both the W’s. Let us consider e+e− → W+W− with both
Pe− Pe+ model √s=300 GeV √s=500 GeV √s=800 GeV √s=1000 GeV
SM 13.598 7.208 3.744 2.692
E6(χ) 13.618 7.227 3.767 2.724
E6(ψ) 13.602 7.212 3.749 2.699
0 0 E6(η) 13.612 7.223 3.762 2.717
LRSM 13.601 7.212 3.750 2.701
ALRSM 13.571 7.183 3.720 2.666
LHM 13.679 7.347 3.867 2.973
SM 53.973 28.716 14.940 10.746
E6(χ) 54.042 28.777 14.994 10.798
E6(ψ) 54.005 28.743 14.963 10.767
-1 1 E6(η) 54.038 28.774 14.991 10.795
LRSM 54.013 28.753 14.981 10.800
ALRSM 53.895 28.640 14.862 10.660
LHM 54.214 29.205 15.362 11.796
SM 41.023 21.826 11.355 8.167
E6(χ) 41.076 21.871 11.396 8.206
E6(ψ) 41.048 21.845 11.372 8.183
-0.9 0.6 E6(η) 41.073 21.869 11.393 8.204
LRSM 41.053 21.850 11.376 8.187
ALRSM 40.964 21.767 11.295 8.102
LHM 41.208 22.198 11.676 8.966
SM 0.418 0.114 0.037 0.023
E6(χ) 0.427 0.122 0.045 0.030
E6(ψ) 0.401 0.103 0.028 0.015
1 -1 E6(η) 0.411 0.110 0.034 0.020
LRSM 0.390 0.093 0.018 0.003
ALRSM 0.390 0.093 0.018 0.003
LHM 0.500 0.182 0.105 0.096
SM 0.857 0.374 0.178 0.125
E6(χ) 0.865 0.381 0.184 0.131
E6(ψ) 0.845 0.365 0.171 0.119
0.9 -0.6 E6(η) 0.853 0.371 0.176 0.123
LRSM 0.837 0.360 0.167 0.115
ALRSM 0.835 0.357 0.162 0.109
LHM 0.922 0.431 0.233 0.191
Table 3: Total cross section (in pb) for different models with both polarized and unpolarized beams. The parameter used for LHM isf=1 TeV andc=0.3, and for theZ′ modelsθ= 0.003 and ∆M = 100 MeV
9
0 200 400 600 800 1000 0
0.16 0.32 0.48 0.64 0.8
sHGeVL
ΣHpbL
700 750 800 850 900 950 0
0.02 0.04 0.06 0.08 0.1
Figure 2: Total cross section forW+ W− production in ane+e− collision for SM[Red-Solid], ALRSM, LRSM [Magenta-Dashed], LHM [Green-Dotted] and E6(χ) [Blue-Dotted] with polarized beams with Pe−=1 and Pe+=-1. ForU(1)′ models, θ= 0.003 and ∆M = 100 MeV are considered. The parameter values off=1 TeV andc=0.3 are used for LHM
W’s decaying into leptons. The differential cross section in this case can be expressed as
dσ
dcos Θdcosθ∗− dφ∗−dcosθ+∗ dφ∗+
= 9β
8192π3sB(W−→l−ν¯l)B(W+→l+νl)Pλλ′λ¯λ¯′Dλλ′ D¯λλ¯¯′, (12) where Θ is the scattering angle andβ =p
(1−4m2W/s) is the velocity of the W in the centre of mass frame,√
sbeing the centre of mass energy. The other angles,θ∗∓ andφ∗∓ are the polar and azimuthal angles of the lepton/antilepton in the rest-mass frame of W∓ with boost direction of W− along the z-axis, respectively. The production and decay tensors,Pλλ′¯λλ¯′, Dλλ′ and ¯Dλ¯λ¯′ are given in [3]. The energy of thel∓ in the centre of mass frame is related to θ∗∓ in the following way:
El∓ =
√s
4 1±βcosθ∗∓
, (13)
This allows us to obtain the double energy differential cross section from Eq. 12 as
dσ dEl−dEl+ =
Z 2 βγmW
2
dσ
dcos Θ dcosθ−∗ dφ∗−dcosθ∗+ dφ∗+
dcos Θdφ∗−dφ∗+. (14)
Following the notation of Dicus and Kallianpur [12], the energies of the leptons El± are expressed as dimensionless variablesXl± defined as:
Xl± = 2 β√
s
El±−
√s 4 (1−β)
. (15)
Xl± varies between 0 and 1. In Table 4 we present the percentage of events in bins ofXl− andXl+ obtained from
1 σ
d2σ dXl+dXl−
at √s = 800 GeV using unpolarized beams. As expected in the case of SM the distribution peaks at maximum values of Xl− and Xl+. This behaviour dominates in other models as well. Since the matrix in Table 4 is symmetric under interchange of Xl− and Xl+ (see Ref. [12] for explicit expression), we have shown only the upper half. We have numerically checked that the matrix is indeed symmetric.
A combination of the effects result in about 5% deviation in the case ofE6(χ) and ALRSM models, but with a qualitative difference. In the case ofE6(χ) it is an enhancement, whereas in the ALRSM case there is a reduction. The effect is reduced to smaller than 4% in the case ofE6(η) model, while in the case of E6(ψ) and LRSM models it is a very negligible contribution of about 1%. The most sensitive bin in all the cases is the one with 0.2≤(Xl−, Xl+)≤0.4, with about 0.6% of the total events in the case of SM. Such small deviation of a few percent is therefore hard to detect even at a high luminosity machine like the ILC. On the other hand, the LHM model, while showing similar qualitative behaviour, deviates from the SM value by about 34% in the same bin, leaving scope of detection at the ILC.
In the above analysis we have not made any attempt to optimize our results by considering different binning options. We expect the qualitative behaviour to remain more or less the same even in the optimal case. This is supported by Fig. 3 and Fig. 4, where we plot the double energy distribution for SM and the deviation from SM expressed as
d2σ dXl−dXl+
model− dXl−d2dXσ l+
SM d2σ
dXl−dXl+
SM
.
We can see from Fig. 3 as discussed before, in the case of SM the distribution peaks at maximum values ofXl− andXl+.
The use of right-handed electron beam and left-handed positron beam is expected to have a much larger sensitivity to all the models. In the case of LHM, while the new gauge boson do not contribute, owing to the fact that this does not couple to the right-handed electrons, the changed fermionic couplings of the SM gauge boson provides substantial effect [10]. In this case, the t- channel contribution is not present, and therefore both the SM as well as new
11
Xl+ 0.2 0.4 0.6 0.8 1.0 Xl−
1.0
0.428 2.705 7.174 13.837 22.693a
0.433 2.707 7.174 13.773 22.566b
0.429 2.705 7.170 13.824 22.665c
0.431 2.707 7.159 13.788 22.594d
0.429 2.706 7.170 13.823 22.664e
0.422 2.702 7.197 13.907 22.832f
0.465 2.716 7.023 13.386 21.804g
0.8
0.355 1.747 4.459 8.488
0.368 1.772 4.474 8.475
0.358 1.753 4.462 8.486
0.365 1.767 4.471 8.478
0.358 1.753 4.462 8.486
0.341 1.720 4.441 8.503
0.450 1.920 4.567 8.389
0.6
0.364 1.053 2.418
0.379 1.087 2.452
0.367 1.060 2.425
0.376 1.080 2.445
0.367 1.060 2.426
0.348 1.016 2.381
0.475 1.293 2.657
0.4
0.454 0.622
0.466 0.652
0.456 0.628
0.463 0.645
0.456 0.629
0.442 0.590
0.541 0.833
0.2
0.626 0.628 0.626 0.628 0.626 0.623 0.646
Table 4: Percentage of events at√s=800 GeV with unpolarized beams in bins ofXl− andXl+corresponding to different models: a=SM,b=E6(χ),c=E6(ψ), d=E6(η),e=LRSM,f=ALRSM andg=LHM. The parameters forU(1)′models,
Figure 3: Double energy distribution for the SM at√
s= 800 GeV with unpo- larized beams
physics contributions show a symmetric behaviour. The difference in the two cases is, as expected, a constant shift, either positive or negative. At√s= 800 GeV, about 50% deviation is seen for both LRSM and ALRSM models, while E6(ψ) has about 40% deviation. The other two models show slightly reduced sensitivity with about 30% and 12% deviations in the case ofE6(χ) andE6(η) models respectively.
3.3 Single Energy Distribution
Energy distribution of the secondary lepton obtained by integrating the double energy distribution in Eqn. 14 overEl+, is another observable that might give
Figure 4: Double energy distribution showing the percentage deviation for the two models (a)E6(χ) and (b) LHM from the SM at√
s= 800 GeV with unpo- larized beams. In the case ofE6(χ)θ= 0.003 and ∆M = 100 MeV, and in the case of LHMf = 1 TeV andc= 0.3 are used
13
a handle on the new effects. In Figs. 5, 6 we plot the energy distribution of the lepton for the case of√s = 800 GeV for both ideal and realistic degrees of beam polarizations. The signs of the beam polarization are chosen so as to essentially switch off the SM t-channel contribution, so as to enhance the effects of the new physics. There is no appreciable deviation in the case of unpolarized beams (and in the case of left-handed electron beams). We notice that the LRSM and ALRSM cases behave qualitatively differently compared to theE6
models and the LHM. While in the former case there is a reducing effect for the entire range ofXl−, the latter has an increasing effect. This could be used as a discriminating factor between the left-right symmetric models and the other models used.
0.0 0.2 0.4 0.6 0.8 1.0
0.000 0.005 0.010 0.015
Xl- dΣdXl-
Figure 5: Energy distribution of the sec- ondary leptons. with polarized beams with Pe−=1 and Pe+=-1 at √
s = 800 GeV.
The parameters used are ∆M=100 MeV andθ=0.003 forU(1)′ type of models, and f = 1 TeV and c = 0.3 in the case of LHM. Different curves correspond to SM (Red-Solid), LHM (Green-Dotted), E6(χ) (Blue-Dotted), and ALRSM and LRSM (Magenta-Dashed).
0.0 0.2 0.4 0.6 0.8 1.0
0.00 0.01 0.02 0.03 0.04
Xl- dΣdXl-
Figure 6: Energy distribution of the sec- ondary leptons. with polarized beams with Pe−=0.9 andPe+=-0.6 at√
s= 800 GeV.
The parameters used are ∆M=100 MeV andθ=0.003 forU(1)′ type of models, and f = 1 TeV and c = 0.3 in the case of LHM. Different curves correspond to SM (Red-Solid), LHM (Green-Dotted), E6(χ) (Blue-Dotted), and ALRSM and LRSM (Magenta-Dashed).
Note that in the case of realistic degrees of polarization since the neutrino t- channel effects are present, the panels comparing ideal and realistic degrees of freedom appear quite different as functions ofXl−.
3.4 Angular Spectrum of the Secondary Lepton
We next consider the angular distribution of one of the secondary leptons. The way of calculating the angular distribution is done in our earlier work [10], which we follow here. The angular distributions for different polarization com- binations in case of different models is calculated for√
s=800 GeV. As in the earlier cases, the case of unpolarized beams and the case of left-handed electron beams are not significant in the case of angular distribution as well. But the
case of right-handed polarization of electron beam along with the left handed polarization of the positron beam provides much better discrimination. This distribution is shown in the Figs. 7, 8 for both the ideal and realistic degrees of beam polarizations. Qualitatively, the picture as regards discrimination be- tween the models, remains more or less the same as in the case of single energy distribution, except that there is a small difference between the two cases of left-right symmetric models considered. In the best case, the ALRSM model shows a deviation of around 45% for the parameter set used.
-1.0 -0.5 0.0 0.5 1.0
0.000 0.002 0.004 0.006 0.008
cosΘl
dΣ dcosΘl
Figure 7: Angular distribution of one of the secondary leptons at √
s=800 GeV using polarized beams with Pe−=1 and Pe+= -1 in the case of SM (Red- Solid), LHM (Green-Dotted),E6(χ) (Blue- Dotted), ALRSM (Yellow-DotDashed) and LRSM (Magenta-Dashed). The parameters used are ∆M=100 MeV andθ = 0.003 in the case ofE6 and LR models, andf = 1 TeV andc= 0.3 in the case of LHM.
-1.0 -0.5 0.0 0.5 1.0
0.000 0.005 0.010 0.015 0.020
cosΘl
dΣ dcosΘl
Figure 8: Angular distribution of one of the secondary leptons at √
s=800 GeV using polarized beams with Pe−=0.9 and Pe+= -0.6 in the case of SM (Red- Solid), LHM (Green-Dotted),E6(χ) (Blue- Dotted), ALRSM (Yellow-DotDashed) and LRSM (Magenta-Dashed). The parameters used are ∆M=100 MeV andθ = 0.003 in the case of E6 and LR models, andf = 1 TeV andc= 0.3 in the case of LHM..
Fig. 8 shows the forward-backward asymmetric behaviour arising from the Zeecoupling, which is different for different models considered. With this obser- vation, we may obtain the fraction of leptons emitted in the backward direction, which may be defined as
fback = R0
−1(dσ/dcosθl) dcosθl
R1
−1(dσ/dcosθl) dcosθl
.
This fraction is a useful quantity to consider in the case of unpolarized and left-polarized electron beams as well. In Table 5 we present these fractions for
√s = 500 GeV, 800 GeV and 1000 GeV. The result shows with unpolarized beams about 3-4 % of the leptons are emitted in the backward hemisphere. The deviation is about 3% in going from SM toE6(χ) and ALRSM which are more sensitive compared to other models. At√s=800 GeV, the deviation becomes more significant and is about 8%. However LHM being more sensitive shows
∼36 % deviation at √s=500 GeV which further increases with energy. This 15
above results are slightly increased by switching on the beam polarization to left-handed electrons and right-handed positrons. The departure from ideal degrees of polarization does not play a significant role at the level of significant places retained in the tables for this configuration.
We see from Table 5 that for the case of polarization with about 90% of right handed electrons and 60% of left handed positrons, the deviation increases.
Owing to the near absence of the dominant t- channel neutrino diagram with this polarization configuration, the new physics contribution shows up in the fraction of the leptons emitted in the backward direction. Thus at √s= 800 GeV for E6(ψ) andE6(χ) there is about 15% deviation from SM, whereasE6(η) is more constrained with only 5% deviation. The symmetric models ALRSM and LRSM are more sensitive with about 30% deviation.
In the case of completely right handed electrons and left handed positrons, from the same table it may be seen that all the models give rise to the same prediction as the SM. This is a consequence of the near symmetric behaviour in the angular distributions as shown in Fig. 7. Due to this property the new physics effects in fback and AF B are essentially wiped out. It is important to note that this feature persists with completely right-polarized electron and partially left-polarized positrons, andvice versa.
Another useful observable related to the angular asymmetry is the forward- backward asymmetry defined as
AF B= R0
−1(dσ/dcosθl)dcosθl−R1
0(dσ/dcosθl)dcosθl
R1
−1(dσ/dcosθl)dcosθl
. (16) The values for this asymmetry is also presented in Table 5 for different models at three different collider energies. This asymmetry shows a similar behaviour as that of the fraction of the leptons emitted in the backward direction as regards the beam polarization configurations and discrimination between the models.
In fact, the two observables are intimately related: AF B = 2fback−1. How- ever, since they are experimentally realized differently, it may be appropriate to consider both of them.
3.5 Left-Right Asymmetry
The left-right asymmetry is another important observable to be considered for studying the new models. We define the left-right asymmetry in the differential cross section as:
Adif fLR = dσ(e+Re−L)/dcosθ−dσ(e+Le−R)/dcosθ
dσ(e+Re−L)/dcosθ+dσ(e+Le−R)/dcosθ, (17) whereθis theW scattering angle.
TheZ′ of theU(1)′ models considered here couples to both left- and right- handed fermions, but with varying relative couplings. Thus, one would expect appreciable change in the asymmetry between the left- and right-polarized cross
√s=500GeV √s=800GeV √s=1000GeV
Pe− Pe+ Model fback AF B fback AF B fback AF B
SM 0.0346 -0.9308 0.0238 -0.9524 0.0208 -0.9585 E6(χ) 0.0356 -0.9288 0.0257 -0.9487 0.0234 -0.9532 E6(ψ) 0.0347 -0.9305 0.0242 -0.9517 0.0213 -0.9574 0 0 E6(η) 0.0353 -0.9293 0.0252 -0.9496 0.0228 -0.9545 LRSM 0.0347 -0.9306 0.0241 -0.9517 0.0213 -0.9573 ALRSM 0.0330 -0.9339 0.0208 -0.9585 0.0161 -0.9677 LHM 0.0474 -0.9053 0.0474 -0.9052 0.0212 -0.9574 SM 0.0324 -0.9353 0.0225 -0.9550 0.0197 -0.9606 E6(χ) 0.0332 -0.9335 0.0241 -0.9518 0.0219 -0.9561 E6(ψ) 0.0327 -0.9345 0.0232 -0.9537 0.0206 -0.9588 -0.9 0.6 E6(η) 0.0332 -0.9336 0.0240 -0.9520 0.0218 -0.9563 LRSM 0.0328 -0.9343 0.0233 -0.9533 0.0209 -0.9583 ALRSM 0.0312 -0.9376 0.0201 -0.9598 0.0160 -0.9681 LHM 0.0440 -0.9120 0.0441 -0.9118 0.0167 -0.9666 SM 0.1618 -0.6765 0.1061 -0.7878 0.0909 -0.8183 E6(χ) 0.1694 -0.6612 0.1208 -0.7583 0.1116 -0.7788 E6(ψ) 0.1518 -0.6964 0.0889 -0.8222 0.0680 -0.8639 0.9 -0.6 E6(η) 0.1587 -0.6826 0.1013 -0.7975 0.0845 -0.8310 LRSM 0.1457 -0.7087 0.0786 -0.8427 0.0552 -0.8896 ALRSM 0.1426 -0.7148 0.0648 -0.8703 0.0268 -0.9463 LHM 0.2174 -0.5653 0.2120 -0.5760 0.2251 -0.5498 1 -1 All models 0.5834 0.1668 0.5392 0.0785 0.5264 0.0528
Table 5: Fraction of leptons emitted in the backward direction, and the forward- backward asymmetry for all models for unpolarized and polarized beams with a parameter choice ofθ=0.003 and ∆M= 0.1 GeV. The parameters for LHM aref=1 TeV andc=0.3
17
sections. But compared to these the new gauge boson in the LHM has the peculiar property of coupling only to left-handed fermions as mentioned earlier.
Fig. 9(a) shows the LR asymmetry for√
s=800 GeV. Even at low energies the deviation becomes apparent. Notice also that the LHM now pair with the left- right symmetric models unlike in the case of energy and angular distributions shown in Figures 5 and 7, providing us another tool for discriminating different models from each other.
@aD
-1.0 -0.5 0.0 0.5 1.0
0.80 0.85 0.90 0.95 1.00
cosΘ
AdiffLR
@bD
0 200 400 600 800 1000
0.98 0.985 0.99 0.995 1
sHGeVL
ALR
Figure 9: (a) Differential Left-right asymmetry as a function of the scattering angle for different models at √s= 800 GeV, and (b) the integrated left-right asymmetry as a function of the centre of mass energy for different models:
SM (Red-Solid), LHM(Green-Dotted), E6(χ) (Blue-Dotted), LRSM(Magenta- Dashed) and ALRSM (Yellow-DotDashed). Parameters used areθ= 0.003 and
∆M = 100 MeV for theE6and LR models, andf = 1 TeV andc= 0.3 for the LHM.
We may go one step further by considering an integral version of this asym- metry as better efficiency may be obtained this way, by integrating each of the differential cross sections from an opening angleθ0up to an angleπ−θ0, for var- ious realistic values ofθ0to which the data can be integrated without difficulty.
We define the integrated left-right asymmetry as:
ALR= σθ0(e+Re−L →W W)−σθ0(e+Le−R→W W)
σθ0(e+Re−L →W W) +σθ0(e+Le−R→W W) (18) whereσθ0 stands forRπ−θ0
θ0 (dσ/dθ) dθ.
This asymmetry, for different parameter models is plotted against the cen- tre of mass energy in Fig. 9(b) with θ0 =0. Dominance of the t-channel in the W-pair production establishes a highly forward peaked cross section, whereas as is seen from Fig. 9(a) the deviations in LR asymmetry grows with the scat- tering angle, except for a region nearθ = 180 degrees. Thus, the cut-off angle
can be effectively used to increase the deviation in the asymmetry. The results are presented in Table 6 for different cut off angles in different models at dif- ferent centre of mass energies. An asymmetric cut off, meaning and integrated asymmetry between scattering angles, θ10 and π−θ20, with θ20 6= θ10, may improve the situation quantitatively. Again, as in the case of other observables, our gaol in this report is to demonstrate the viability of identifying deviations from the SM case, and further providing tools to disentangle different possible models. With this illustrative purpose in mind, we do not attempt to optimize the investigation.
4 Discussion and Conclusions
In the present work we have considered a class of additionalZ′models which are of interest to the linear collider community, both at the ILC as well as at CLIC.
While the masses of these bosons are already required to be significantly high, their imprint through mixing with the standard modelZ boson is the subject of this investigation. By considering popularE6 and left-right symmetric models like LRSM and ALRSM we have demonstrated that the new physics signatures due to these models can be imprinted only at higher center of mass energies.
In the LHM model considered in our earlier work [10], Z′ had the property of coupling only to left-handed fermions. This is in contrast to the other models considered here whereZ′behaves like the SMZ. Thus compared to LHM, these models do not show appreciable deviation at lower center of mass energies.
While our focus inW-pair production at ILC is on the unambiguous signal that it provides forZSM −Z′ mixing, we notice the interesting possibility of model discrimination here. For example, let us compare the deviations of some of the observables from their SM values. In the case of energy and angular momentum distributions notice the qualitatively different behaviour of left-right symmetric models compared to E6 models and LHM. Similarly, in the case of integrated and differential left-right asymmetries theE6 model has a different qualitative behaviour compared to the LHM and left-right symmetric models.
Note that W W Zi (i = 1,2) coupling is insensitive to differences between Z′ models, and therefore it is the coupling of Z′ with the initial electron and positron that enables any possible model discrimination in the present case.
Thus, one would imagine that fermion pair production process is better suited to distinguish different models. In a recent study, the potential of the ILC to discriminate betweenZ′ models through fermion pair production is studied in Ref. [16]. Here the sensitivity is studied by considering cross sections which are shown to be sensitive to new physics due to the availability of high statistics at the ILC. It is also shown that beam polarization does enhance the sensitivity in an essential manner. Similar studies with Drell-Yan dilepton production [17] also probe distinguishability of differentZ′ models at LHC. At the same time, our study indicates thatW-pair production at ILC is capable of supplementing the fermion pair production process in model discrimination in the case ofZ′models.
More extensive analysis involving a parameter scan, also incorporating a realistic
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ALR
Cut off angleθ0 Model √
s=500GeV √
s= 800 GeV √
s= 1000 GeV
SM 0.9921 0.9950 0.9958
E6(χ) 0.9914 0.9936 0.9930
E6(ψ) 0.9930 0.9967 0.9984
0 E6(η) 0.9924 0.9956 0.9968
LRSM 0.9935 0.9976 0.9995
ALRSM 0.9935 0.9976 0.9995
LHM 0.9945 0.9972 0.9983
SM 0.9853 0.9872 0.9877
E6(χ) 0.9842 0.9836 0.9800
E6(ψ) 0.9870 0.9916 0.9955
15 E6(η) 0.9860 0.9889 0.9909
LRSM 0.988 0.9939 0.9985
ALRSM 0.9879 0.9938 0.9985
LHM 0.9885 0.9933 0.9956
SM 0.9742 0.9765 0.9771
E6(χ) 0.9723 0.9703 0.9638
E6(ψ) 0.9772 0.9846 0.9917
30 E6(η) 0.9755 0.9797 0.9834
LRSM 0.979 0.9889 0.9973
ALRSM 0.9787 0.9885 0.9971
LHM 0.9808 0.9890 0.9928
SM 0.9601 0.9622 0.9628
E6(χ) 0.9574 0.9531 0.9434
E6(ψ) 0.9648 0.9753 0.9866
45 E6(η) 0.9622 0.9679 0.9738
LRSM 0.9676 0.9822 0.9956
ALRSM 0.9669 0.9811 0.9951
LHM 0.9721 0.9845 0.9899
SM 0.9453 0.9466 0.9470
E6(χ) 0.9418 0.9348 0.9224
E6(ψ) 0.9517 0.9652 0.9811
60 E6(η) 0.9483 0.9552 0.9638
LRSM 0.9555 0.9750 0.9939
ALRSM 0.9542 0.9727 0.9927
LHM 0.9640 0.9807 0.9875
Table 6: Left-right asymmetry for different cut-off angles at selected√svalues for different models considered with a parameter choice ofθ=0.003 and ∆M= 0.1 GeV. The parameters for LHM aref=1 TeV andc=0.3
collider-detector simulation is required to draw conclusions regarding ability of W-pair production in model discrimination at ILC, adapting the procedure in Ref. [16]. Alternatively one may contemplate adapting different asymmetries of the type we have proposed for fermion pair production as well. Perhaps a joint analysis involving both fermion pair production andW-pair production is needed to bring out the full potential of ILC in this regard.
Our conclusions for our models is as follows: we have studied in detail each of the E6 models denoted by χ, ψ and η and the LRSM and ALRSM models.
We find that at a centre of mass energy of 800 GeV, all these models show a sharp deviation from the SM predictions for energy-energy, single-energy and angular correlations of the decay lepton(s) generally for right-handed electron and left-handed positron with realistic degrees of polarization. Curiously for ideal polarization the decay lepton fraction and AF B are insensitive to new physics for the same configuration above, as the t- channel contribution is totally absent in this case. Theχ, LRSM and ALRSM models are more sensitive than the other models even with unpolarized beams. The reason for the absence of sensitivity at lower energies is due to the stringent bounds already present on the parameters of the models. The FB and LR asymmetries have also been studied, with the latter being more sensitive to new physics in all the models.
These models remain less sensitive than the LHM model studied earlier. Any indication ofU(1)′ type ofZ′ in W-pair production at ILC is a clear signal of ZSM −Z′ mixing. Apart from this, the present study points to the potential of this process to supplement the fermion pair production process in model discrimination, through a suitable combination of observables considered.
Overall a strong polarization program at the ILC will significantly enhance the diagnostic capability towards additional Z′ bosons particularly at higher design energies.
Acknowledgments: BA thanks the Department of Science and Technology, Government of India and the Homi Bhabha Fellowships Council for support during the course of these investigations. MP thanks the Department of Physics, Indian Institute of Technology Guwahati for hospitality when part of this work was done. PP thanks BRNS, DAE, Government of India for support through a project (No.: 2010/37P/49/BRNS/1446).
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